#355853

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1241 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,17)(8,23,27,14)(9,10)(11,15)(12,16)(13,20)(18,31,34,33)(19,22)(21,30)(24,29,35,36)(25,28,32,26);(1,2,3,4)(5,7,11,21,12,22,10,20)(6,13,9,19,16,30,15,17)(8,24,28,18,27,35,26,34)(14,31,32,29,23,33,25,36);(5,8)(6,14)(7,18)(9,25)(10,28)(11,26)(12,27)(13,29)(15,32)(16,23)(17,33)(19,31)(20,35)(21,24)(22,34)(30,36);(5,9,11,16,12,15,10,6)(7,19,21,30,22,17,20,13)(8,25,26,23,27,32,28,14)(18,31,24,36,34,33,35,29)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=egi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,h^a=hi,c^b=ce,e^c=ei,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226669, #72226671, #72226674
maximal normal subgroup of	#72226669, #72226671, #72226674

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263049);